Integrand size = 37, antiderivative size = 101 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}+\frac {e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 44, 65, 214} \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}}-\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \]
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Rule 44
Rule 65
Rule 214
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x)^2 \sqrt {d+e x}} \, dx \\ & = -\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {e \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 \left (c d^2-a e^2\right )} \\ & = -\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{c d^2-a e^2} \\ & = -\frac {\sqrt {d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)}+\frac {e \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (c d^2-a e^2\right )^{3/2}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d+e x}}{\left (-c d^2+a e^2\right ) (a e+c d x)}+\frac {e \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\sqrt {c} \sqrt {d} \left (-c d^2+a e^2\right )^{3/2}} \]
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Time = 27.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {e x +d}}{c d x +a e}+\frac {e \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{e^{2} a -c \,d^{2}}\) | \(82\) |
derivativedivides | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (e^{2} a -c \,d^{2}\right ) \left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )}+\frac {\arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \left (e^{2} a -c \,d^{2}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )\) | \(111\) |
default | \(2 e \left (\frac {\sqrt {e x +d}}{2 \left (e^{2} a -c \,d^{2}\right ) \left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )}+\frac {\arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \left (e^{2} a -c \,d^{2}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )\) | \(111\) |
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Time = 0.36 (sec) , antiderivative size = 353, normalized size of antiderivative = 3.50 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [-\frac {\sqrt {c^{2} d^{3} - a c d e^{2}} {\left (c d e x + a e^{2}\right )} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {c^{2} d^{3} - a c d e^{2}} \sqrt {e x + d}}{c d x + a e}\right ) + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x\right )}}, -\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d e x + a e^{2}\right )} \arctan \left (\frac {\sqrt {-c^{2} d^{3} + a c d e^{2}} \sqrt {e x + d}}{c d e x + c d^{2}}\right ) + {\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt {e x + d}}{a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {e \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} {\left (c d^{2} - a e^{2}\right )}} - \frac {\sqrt {e x + d} e}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} {\left (c d^{2} - a e^{2}\right )}} \]
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Time = 9.75 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {e\,\mathrm {atan}\left (\frac {c\,d\,\sqrt {d+e\,x}}{\sqrt {c\,d}\,\sqrt {a\,e^2-c\,d^2}}\right )}{\sqrt {c\,d}\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}+\frac {e\,\sqrt {d+e\,x}}{\left (a\,e^2-c\,d^2\right )\,\left (a\,e^2-c\,d^2+c\,d\,\left (d+e\,x\right )\right )} \]
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